# Microstrip (Derivation of Capacitance and Inductance)

1. May 3, 2012

### shpongle

Could anyone help me to find out a book which has detailed explaination(derivations) for the capacitance, inductance of a mictropstrip line. It would be helpful if the book has detailed discusions on fringing fields,(effective width of the ground plane) effective permitivty etc.

I have tried out Engineering Electromagnetics by Hayt but it wasn't very helpful.

Thanks

Last edited: May 3, 2012
2. May 3, 2012

### Floid

Introduction to Electromagnetic Compatibility by Clayton Paul has a chapter (4) that discusses it in some detail.

But to paraphrase a quote from that book: "generally per unit length paramaters cannot be found in formula form but are obtained in approximate relations through means like conformal mapping and numerical methods".

You might want to check it out.

He quotes several other sources such as:

H.R. Kaupp, Characteristics of microstrip transmission lines, IEE Trans. Electron. Comput. EC-16, 186-193 (April 1967)
H. A. Wheeler, Transmission line properties of parallel strips sperated by a dielectric sheet, IEEE Trans. Microwave Theory Tech. MTT-13, 172-185 (1965)

3. May 3, 2012

4. May 3, 2012

### shpongle

Thank you for quick reply. I have attached a file. Could you check slide number 8 and 16 and help me to find out a book which derives/explains a similar expression.

Because, the books suggested by you have inductance and capacitance in terms of permitivity, permeability and velocity of light.

While I was looking for inductance and capacitance in term of width and height of the microstrip!!

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5. May 4, 2012

### Floid

It looks like they are just using basic electromagnetic equations. Slide 8 is just the capacitance of a parallel plate capacitor modified because the plates are not of equal size (the problem states to assume a ground plane between them).

In a capacitor the number of charges on the top plate equals the number of charges on the bottom plate. So what it looks like they do to conserve this is form a ratio of the top w^2 to how it would spread out on the bottom (w+h)^2.

On slide 16, the first equation is effective relative permittivity which accounts for the fact that part of the electric field is in the dielectric substrate and part of it is in air. As far as I know this equation was developed empirically and cannot be derived.

The rest of it is just using the permittivity to already known equations.

6. May 4, 2012

### f95toli

What floid said...

Also, I think you need to explain what you are trying to do.
None of those equations will be good enough if you are actually designing something since they are very approximate, then you will need numerical methods (try a free program called TXline) OR you can simply look up experimental values from a suitable reference.

7. May 4, 2012

### marcusl

Your link uses approximations, which were popular before computers became ubiquitous. You can find more extensive compilations of approximations, covering lines, couplers, right angle bends, etc., in older books like Edwards, Foundations for Microstrip Design (1975).

R. Collin derives stripline and simple microstrip geometry (homogeneous dielectric) exactly in his book Field Theory of Guided Waves, 1991. See the chapter on Transmission Lines. Even and odd mode currents are derived as well.

As has been said already, exact results for practical cases (which can include vias, components, etc.) are usually obtained by modeling with, eg., HFSS or VSS.

Last edited: May 4, 2012
8. May 4, 2012

### shpongle

Thank you very much for your help.